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```MEN*  OF MATHEMATICS
course that in the algebraic number field concerned the funda-
mental theorem of arithmetic must hold. After several exciting
but premature communications to the French Academy of
Sciences, he admitted his error. Being restlessly interested in a
large number of other problems at the time, Cauchy turned
aside and failed to make the great discovery which was well
within the capabilities of his prolific genius and left the field to
Kummer. The central difficulty was serious: here was a species
of 'integers' - those of the field concerned - which defied the
fundamental theorem of arithmetic; how reduce them to law
and order?
The solution of this problem by the invention of a totally
new kind of 'number' appropriate to the situation, which (in
terms of these 'numbers') automatically restored the funda-
mental theorem of arithmetic, ranks with the creation of non-
Euclidean geometry as one of the outstanding scientific achieve-
ments of the nineteenth century, and it is well up in the high
mathematical achievements of all history. The creation of the
new 'numbers' - so-called 'ideal numbers' - was the invention
of Kummer in 1845. These new 'numbers' were not constructed
for all algebraic number fields but only for those fields arising
from the division of the circle.
Kummer too had fallen foul of the net which snared Cauchy,
and for a tune he believed that he had proved Fermat's 'Last
Theorem'. Then Dirichlet, to whom the supposed proof was
submitted for criticism, pointed out by means of an example
that the fundamental theorem of arithmetic, contrary to
Rummer's tacit assumption, does not hold in the field con-
cerned. This failure of Kummer's was one of the most fortunate
things that ever happened in mathematics. Like Abel's initial
mistake in the matter of the general quintic, Kummer's turned
him into the right track, and he invented his 'ideal numbers'.
Kummer, Kronecker, and Dedekind, in their invention of the
modern theory of algebraic numbers, by enlarging the scope of
arithmetic ad infinitum and bringing algebraic equations within
the purview of number, did for the higher arithmetic and the
theory of algebraic equations what Gauss, Lobatchewsky,
Johann Bolyai, and Biemann did for geometry in emancipating
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